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Trying prove infinite delta by equiv-reasoning
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Tue, 25 Nov 2014 12:34:09 +0900
parents 73bb981cb1c6
children 474ed34e4f02
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5ba82f107a95 Define Similar in Agda
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1 open import list
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6e6d646d7722 Split basic functions to file
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2 open import basic
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e0ba1bf564dd Apply level to some functions
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3
e0ba1bf564dd Apply level to some functions
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4 open import Level
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742e62fc63e4 Define Monad-law 1-4
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5 open import Relation.Binary.PropositionalEquality
742e62fc63e4 Define Monad-law 1-4
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6 open ≡-Reasoning
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5ba82f107a95 Define Similar in Agda
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8 module delta where
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11 data Delta {l : Level} (A : Set l) : (Set (suc l)) where
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12 mono : A -> Delta A
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13 delta : A -> Delta A -> Delta A
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14
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15 deltaAppend : {l : Level} {A : Set l} -> Delta A -> Delta A -> Delta A
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16 deltaAppend (mono x) d = delta x d
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17 deltaAppend (delta x d) ds = delta x (deltaAppend d ds)
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18
9bb7c9bee94f Trying redefine delta for infinite changes
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19 headDelta : {l : Level} {A : Set l} -> Delta A -> Delta A
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20 headDelta (mono x) = mono x
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21 headDelta (delta x _) = mono x
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23 tailDelta : {l : Level} {A : Set l} -> Delta A -> Delta A
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24 tailDelta (mono x) = mono x
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25 tailDelta (delta _ d) = d
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5ba82f107a95 Define Similar in Agda
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6ce83b2c9e59 Proof Functor-laws
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27
6ce83b2c9e59 Proof Functor-laws
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28 -- Functor
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29 fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> (Delta A) -> (Delta B)
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30 fmap f (mono x) = mono (f x)
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31 fmap f (delta x d) = delta (f x) (fmap f d)
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35 -- Monad (Category)
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36 eta : {l : Level} {A : Set l} -> A -> Delta A
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37 eta x = mono x
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742e62fc63e4 Define Monad-law 1-4
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38
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46b15f368905 Define bind and mu for Infinite Delta
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39 bind : {l ll : Level} {A : Set l} {B : Set ll} -> (Delta A) -> (A -> Delta B) -> Delta B
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40 bind (mono x) f = f x
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41 bind (delta x d) f = deltaAppend (headDelta (f x)) (bind d (tailDelta ∙ f))
46b15f368905 Define bind and mu for Infinite Delta
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46b15f368905 Define bind and mu for Infinite Delta
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43 mu : {l : Level} {A : Set l} -> Delta (Delta A) -> Delta A
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44 mu d = bind d id
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46b15f368905 Define bind and mu for Infinite Delta
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47 returnS : {l : Level} {A : Set l} -> A -> Delta A
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48 returnS x = mono x
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49
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50 returnSS : {l : Level} {A : Set l} -> A -> A -> Delta A
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51 returnSS x y = deltaAppend (returnS x) (returnS y)
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52
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0bc402f970b3 Proof Monad-law 1
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53
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54 -- Monad (Haskell)
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55 return : {l : Level} {A : Set l} -> A -> Delta A
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56 return = eta
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57
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23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
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58 _>>=_ : {l ll : Level} {A : Set l} {B : Set ll} ->
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59 (x : Delta A) -> (f : A -> (Delta B)) -> (Delta B)
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60 (mono x) >>= f = f x
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61 (delta x d) >>= f = deltaAppend (headDelta (f x)) (d >>= (tailDelta ∙ f))
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a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
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6ce83b2c9e59 Proof Functor-laws
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64
6ce83b2c9e59 Proof Functor-laws
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65 -- proofs
6ce83b2c9e59 Proof Functor-laws
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67 -- sub proofs
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68
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69 head-delta-natural-transformation : {l ll : Level} {A : Set l} {B : Set ll} ->
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70 (f : A -> B) (d : Delta A) -> (headDelta (fmap f d)) ≡ fmap f (headDelta d)
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71 head-delta-natural-transformation f (mono x) = refl
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72 head-delta-natural-transformation f (delta x d) = refl
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73
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74 tail-delta-natural-transfomation : {l ll : Level} {A : Set l} {B : Set ll} ->
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75 (f : A -> B) (d : Delta A) -> (tailDelta (fmap f d)) ≡ fmap f (tailDelta d)
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76 tail-delta-natural-transfomation f (mono x) = refl
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77 tail-delta-natural-transfomation f (delta x d) = refl
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78
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79 delta-append-natural-transfomation : {l ll : Level} {A : Set l} {B : Set ll} ->
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80 (f : A -> B) (d : Delta A) (dd : Delta A) ->
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81 deltaAppend (fmap f d) (fmap f dd) ≡ fmap f (deltaAppend d dd)
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82 delta-append-natural-transfomation f (mono x) dd = refl
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83 delta-append-natural-transfomation f (delta x d) dd = begin
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84 deltaAppend (fmap f (delta x d)) (fmap f dd)
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85 ≡⟨ refl ⟩
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86 deltaAppend (delta (f x) (fmap f d)) (fmap f dd)
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87 ≡⟨ refl ⟩
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88 delta (f x) (deltaAppend (fmap f d) (fmap f dd))
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89 ≡⟨ cong (\d -> delta (f x) d) (delta-append-natural-transfomation f d dd) ⟩
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90 delta (f x) (fmap f (deltaAppend d dd))
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91 ≡⟨ refl ⟩
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92 fmap f (deltaAppend (delta x d) dd)
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93 ∎
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94
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95
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96 mu-head-delta : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> mu (headDelta d) ≡ headDelta (mu d)
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97 mu-head-delta (mono (mono x)) = refl
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98 mu-head-delta (mono (delta x (mono xx))) = begin
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99 mu (headDelta (mono (delta x (mono xx))))
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100 ≡⟨ refl ⟩
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101 bind (headDelta (mono (delta x (mono xx)))) id
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102 ≡⟨ refl ⟩
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103 bind (delta x (mono xx)) return
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104 ≡⟨ refl ⟩
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105 deltaAppend (headDelta (return x)) (bind (mono xx) (tailDelta ∙ return))
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106 ≡⟨ refl ⟩
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107 deltaAppend (headDelta (return x)) ((tailDelta ∙ return) xx)
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108 ≡⟨ refl ⟩
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109 deltaAppend (headDelta (mono x)) (tailDelta (mono xx))
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110 ≡⟨ refl ⟩
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111 deltaAppend (mono x) (mono xx)
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112 ≡⟨ refl ⟩
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113 delta x (mono xx)
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114 ≡⟨ {!!} ⟩
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diff changeset
115 headDelta (delta x (mono xx))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
116 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
117 headDelta (bind (mono (delta x (mono xx))) id)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
118 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
119 headDelta (mu (mono (delta x (mono xx))))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
120 ∎
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
121 mu-head-delta (mono (delta x (delta x₁ d))) = {!!}
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
122 mu-head-delta (delta d dd) = {!!}
38
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
123
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
124 -- Functor-laws
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
125
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
126 -- Functor-law-1 : T(id) = id'
55
9c8c09334e32 Redefine Delta for infinite changes in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 54
diff changeset
127 functor-law-1 : {l : Level} {A : Set l} -> (d : Delta A) -> (fmap id) d ≡ id d
57
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
128 functor-law-1 (mono x) = refl
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
129 functor-law-1 (delta x d) = cong (delta x) (functor-law-1 d)
38
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
130
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
131 -- Functor-law-2 : T(f . g) = T(f) . T(g)
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
132 functor-law-2 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
55
9c8c09334e32 Redefine Delta for infinite changes in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 54
diff changeset
133 (f : B -> C) -> (g : A -> B) -> (d : Delta A) ->
9c8c09334e32 Redefine Delta for infinite changes in Agda
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 54
diff changeset
134 (fmap (f ∙ g)) d ≡ ((fmap f) ∙ (fmap g)) d
57
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
135 functor-law-2 f g (mono x) = refl
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
136 functor-law-2 f g (delta x d) = cong (delta (f (g x))) (functor-law-2 f g d)
38
6ce83b2c9e59 Proof Functor-laws
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 36
diff changeset
137
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
138 -- Monad-laws (Category)
62
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
139 {-
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
140 -- monad-law-1 : join . fmap join = join . join
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
141 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d)
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
142 monad-law-1 (mono d) = refl
62
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
143 monad-law-1 (delta x (mono d)) = begin
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
144 (mu ∙ fmap mu) (delta x (mono d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
145 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
146 mu (fmap mu (delta x (mono d)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
147 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
148 mu (delta (mu x) (mono (mu d)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
149 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
150 bind (delta (mu x) (mono (mu d))) id
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
151 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
152 deltaAppend (headDelta (mu x)) (bind (mono (mu d)) tailDelta)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
153 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
154 deltaAppend (headDelta (mu x)) (tailDelta (mu d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
155 ≡⟨ cong (\de -> deltaAppend de (tailDelta (mu d))) (head-delta-natural-transformation {!!} {!!}) ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
156 deltaAppend (mu (headDelta x)) (tailDelta (mu d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
157 ≡⟨ {!!} ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
158 (mu ∙ mu) (delta x (mono d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
159 ∎
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
160 monad-law-1 (delta x (delta d d₁)) = {!!}
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
161 -}
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
162
29
e0ba1bf564dd Apply level to some functions
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
163
62
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
164 {-
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
165 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
166 monad-law-1 (mono d) = refl
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
167 monad-law-1 (delta x (mono d)) = begin
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
168 (mu ∙ fmap mu) (delta x (mono d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
169 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
170 mu ((fmap mu) (delta x (mono d)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
171 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
172 mu (delta (mu x) (fmap mu (mono d)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
173 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
174 mu (delta (mu x) (fmap mu (mono d)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
175 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
176 mu (delta (mu x) (mono (mu d)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
177 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
178 bind (delta (mu x) (mono (mu d))) id
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
179 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
180 deltaAppend (headDelta (mu x)) (bind (mono (mu d)) (tailDelta ∙ id))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
181 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
182 deltaAppend (headDelta (mu x)) (bind (mono (mu d)) (tailDelta))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
183 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
184 deltaAppend (headDelta (mu x)) (tailDelta (mu d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
185 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
186 deltaAppend (headDelta (mu x)) ((tailDelta ∙ mu) d)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
187 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
188 deltaAppend (headDelta (mu x)) (bind (mono d) (tailDelta ∙ mu))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
189 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
190 bind (delta x (mono d)) mu
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
191 ≡⟨ {!!} ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
192 mu (deltaAppend (headDelta x) (tailDelta d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
193 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
194 mu (deltaAppend (headDelta x) (bind (mono d) tailDelta))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
195 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
196 mu (deltaAppend (headDelta (id x)) (bind (mono d) (tailDelta ∙ id)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
197 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
198 mu (deltaAppend (headDelta x) (bind (mono d) (tailDelta ∙ id)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
199 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
200 mu (bind (delta x (mono d)) id)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
201 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
202 mu (deltaAppend (headDelta (id x)) (bind (mono d) (tailDelta ∙ id)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
203 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
204 mu (mu (delta x (mono d)))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
205 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
206 (mu ∙ mu) (delta x (mono d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
207 ∎
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
208 monad-law-1 (delta x (delta xx d)) = {!!}
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
209 {-
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
210 monad-law-1 (delta x d) = begin
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
211 (mu ∙ fmap mu) (delta x d)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
212 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
213 mu ((fmap mu) (delta x d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
214 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
215 mu (delta (mu x) (fmap mu d))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
216 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
217 bind (delta (mu x) (fmap mu d)) id
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
218 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
219 deltaAppend (headDelta (mu x)) (bind (fmap mu d) (tailDelta ∙ id))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
220 ≡⟨ refl ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
221 deltaAppend (headDelta (mu x)) (bind (fmap mu d) (tailDelta ∙ id))
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
222 ≡⟨ {!!} ⟩
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
223 (mu ∙ mu) (delta x d)
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
224 ∎
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
225 -}
0f308ddd6136 Trying prove infinite delta by equiv-reasoning
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 60
diff changeset
226 -}
34
b7c4e6276bcf Proof Monad-law-2-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 33
diff changeset
227
56
bfb6be9a689d Trying redefine monad-laws-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 55
diff changeset
228 {-
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
229 -- monad-law-2-2 : join . return = id
43
90b171e3a73e Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
230 monad-law-2-2 : {l : Level} {A : Set l } -> (s : Delta A) -> (mu ∙ eta) s ≡ id s
35
c5cdbedc68ad Proof Monad-law-2-2
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
231 monad-law-2-2 (similar lx x ly y) = refl
c5cdbedc68ad Proof Monad-law-2-2
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
232
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
233 -- monad-law-3 : return . f = fmap f . return
40
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
234 monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (eta ∙ f) x ≡ (fmap f ∙ eta) x
36
169ec60fcd36 Proof Monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 35
diff changeset
235 monad-law-3 f x = refl
27
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
236
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
237 -- monad-law-4 : join . fmap (fmap f) = fmap f . join
43
90b171e3a73e Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
238 monad-law-4 : {l ll : Level} {A : Set l} {B : Set ll} (f : A -> B) (s : Delta (Delta A)) ->
36
169ec60fcd36 Proof Monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 35
diff changeset
239 (mu ∙ fmap (fmap f)) s ≡ (fmap f ∙ mu) s
40
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
240 monad-law-4 f (similar lx (similar llx x _ _) ly (similar _ _ lly y)) = refl
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
241
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
242
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
243 -- Monad-laws (Haskell)
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
244 -- monad-law-h-1 : return a >>= k = k a
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
245 monad-law-h-1 : {l ll : Level} {A : Set l} {B : Set ll} ->
43
90b171e3a73e Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
246 (a : A) -> (k : A -> (Delta B)) ->
40
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
247 (return a >>= k) ≡ (k a)
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
248 monad-law-h-1 a k = refl
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
249
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
250
40
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
251
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
252 -- monad-law-h-2 : m >>= return = m
43
90b171e3a73e Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
253 monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= return) ≡ m
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
254 monad-law-h-2 (mono x) = refl
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
255 monad-law-h-2 (delta x d) = cong (delta x) (monad-law-h-2 d)
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
256
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
257
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
258
41
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
259
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
260 -- monad-law-h-3 : m >>= (\x -> k x >>= h) = (m >>= k) >>= h
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
261 monad-law-h-3 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
43
90b171e3a73e Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
262 (m : Delta A) -> (k : A -> (Delta B)) -> (h : B -> (Delta C)) ->
41
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
263 (m >>= (\x -> k x >>= h)) ≡ ((m >>= k) >>= h)
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
264 monad-law-h-3 (mono x) k h = refl
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
265 monad-law-h-3 (delta x d) k h = begin
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
266 (delta x d) >>= (\x -> k x >>= h)
42
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
267 ≡⟨ refl ⟩
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
268 -- (delta x d) >>= f = deltaAppend (headDelta (f x)) (d >>= (tailDelta ∙ f))
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
269 deltaAppend (headDelta ((\x -> k x >>= h) x)) (d >>= (tailDelta ∙ (\x -> k x >>= h)))
42
1df4f9d88025 Proof Monad-law-3 (haskell)
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 41
diff changeset
270 ≡⟨ refl ⟩
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
271 deltaAppend (headDelta (k x >>= h)) (d >>= (tailDelta ∙ (\x -> k x >>= h)))
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
272 ≡⟨ {!!} ⟩
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
273 ((delta x d) >>= k) >>= h
41
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
274 ∎
57
dfcd72dc697e ReDefine Delta used non-empty-list for infinite changes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 56
diff changeset
275 -}